On best rational approximation of analytic functions

This paper contains some theorems related to the best approximation ρ_n(f;E) to a function f in the uniform metric on a compact set by rational functions of degree at most n. We obtain results characterizing the relationship between ρ_n(f;K) and ρ_n(f;E) in the case when complements of compact sets K and E are connected, K is a subset of the interior Ω of E, and f is analytic in Ω and continuous on E.     

Best uniform approximation to a class of rational functions

We explicitly determine the best uniform polynomial approximation to a class of rational functions of the form 1/²(x−c)+K(a,b,c,n)/(x−c) on [a,b] represented by their Chebyshev expansion, where a, b, and c are real numbers, n−1 denotes the degree of the best approximating polynomial, and K is a constant determined by a, b, c, and n. Our result is based on the explicit determination of a phase angle η in the representation of the approximation error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best approximating polynomial to a function represented by its Chebyshev expansion. Combined with the phase angle method, this ansatz can be used to find the best uniform approximation to some more functions.     

On the approximation by rational combinations of a class of functions

() [0,1] — {(n)} — , +. , f(x) [0,1] () , x ₁ ,x ₂ [0, 1], (₁)=(₂), f(x ₁ )=f(x ₂ ).     

On the rate of convergence of the partial sums of dirichlet series and rational approximation of analytic functions

We study the asymptotic behavior of the rate of convergence of Dirichlet series that are absolutely convergent in a half-plane; the results obtained are applicable to rational approximation of functions analytic in the unit disk with nonnegative Taylor coefficients. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 116–122.     

On the rate of rational approximation of functions on tangent continua

Upper and lower bounds are established for the rate of rational approximation of functions piecewise analytic on tangent continua. In some special cases, these bounds are coordinated depending on the mutual location of the continua.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1522–1533, November, 1993.     

Best approximation of the differentiation operator on the class of functions analytic in a strip

On the class of functions analytic in a strip, we study a number of related extremal problems for the differentiation operator: the best approximation of the operator, computation of its modulus of continuity, and optimal recovery of the operator from boundary values of a function given with error on a straight line.     

Rational approximation of analytic functions

The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Padé approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Padé approximants, to the application of multipoint Padé approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Padé-Hermite approximation for systems of Markov type functions.     

A simple method of rational approximation of functions

It is shown that it is possible to apply a method proposed by V. K. Dzyadyk for the construction of rational functions that realize a near optimal approximation of entire elementary functions. The method may be termed linear in the sense that all the coefficients of the rational functions are determined from two systems of linear algebraic equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 998–1000, July, 1992.     

Optimal rate of integration and ɛ-entropy of a class of analytic functions

The author considers a class F of analytic functions real in the interval [-1, 1] and bounded in the unit circle. As an estimate of the optimal quadrature error R(n) over the class F it is shown that $$_e - \left( {2\sqrt 2 + \frac{1}{{\sqrt 2 }}} \right)\pi \sqrt n \leqslant R(n) \leqslant e^{ - \frac{\pi }{{\sqrt 2 }}n} .$$ With the additional condition that \(\mathop {max}\limits_{x \in [ - 1,1]}\) ¦f(x)¦?B, an estimate is obtained for the ?-entropy H_?(F): $$\frac{8}{{27}}\frac{{(1n2)^2 }}{{\pi ^2 }} \leqslant \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{H_\varepsilon (F)}}{{\left( {\log \frac{1}{\varepsilon }} \right)^3 }} \leqslant \frac{2}{{\pi ^2 }}(1n2)^2 .$$      

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

Uniform approximation of continuous functions by rational functions

Summary Viene data la condizione necessaria e sufficiente perchè le funzioni razionali di una variabile, aventi poli di ordine prefissato in assegnati punti del piano complesso, costituiscano un sistema completo iu C^o (0, 1). A Bruno Finzi nel suo70 ^mo compleanno. This research has been sponsored in part by the Aerospace Research Laboratoires through the European Office of Aerospace Research, OAR, United States Air Force, under Grant EOOAR-69-0066. Entrata in Redazione il 22 aprile 1970.     

关于一类解析函数的性质

引进用Hλ算子定义的一类解析函数Pλ(μ,α,β).我们导出该类中函数的积分表达式,证明偏差定理,并推广了文[3]中的主要结果.同时改进了[4]中的一个不等式.